%I #13 Oct 23 2022 01:05:38
%S 1,2,4,6,3,9,12,8,10,5,15,18,14,7,21,24,16,20,25,30,22,11,33,27,36,26,
%T 13,39,42,28,56,70,35,105,45,60,32,34,17,51,48,38,19,57,54,40,50,80,
%U 90,63,49,77,44,46,23,69,66,52,65,55,88,58,29,87,72,62,31,93
%N a(1)=1, a(2)=2. Thereafter, if there are prime divisors p of a(n-1) which do not divide a(n-2), a(n) is the least novel multiple of any such p. Otherwise a(n) is the least novel multiple of the squarefree kernel of a(n-1). See comments.
%C In other words, if a(n-1) has k prime divisors p_j, 1 <= j <= k which do not divide a(n-2), where 1 <= k <= omega(a(n-1)), and if m_j*p_j is the least multiple of p_j which is not already a term, then a(n) = Min{m_j*p_j; 1<=j<=k}. Otherwise every prime divisor of a(n-1) also divides a(n-2), in which case a(n) is the least multiple of the squarefree kernel of a(n-1) which is not already a term. If a(n-1) and a(n-2) are coprime the computation of a(n) ranges over all prime divisors of a(n-1). This happens only once (n=3), after which all adjacent terms share a common divisor (as in the EKG sequence, A064413).
%C Departs from A064413 and A352187 at a(19), a(31) respectively, and apparently shares the way odd primes are proven to appear in the former and conjectured to appear in the latter; namely as 2*p, p, 3*p.
%C Conjectures: Permutation of the positive integers with primes in natural order, appearing in same way as in EKG.
%C From _Michael De Vlieger_, Oct 22 2022 (Start)
%C An algorithm similar to the Rains algorithm for the EKG sequence efficiently generates the sequence.
%C Like the EKG sequence, this sequence forces primes into divisibility; Primes divide their predecessors and successors. Consequently they exhibit Lagarias-Rains-Sloane chain 2p -> p -> 3p outside of p = 2, just as in the EKG sequence.
%C Let us define several quasi-rays conspicuous in the scatterplot. From lowest to highest apparent slope, we have the following:
%C - beta: local minima, i.e., a(1)=1 and primes p in order.
%C - gamma: 2p, 4p, and certain other composites.
%C - alpha-k: k*p from large k to k = 3. This system appears as a series of fine quasi-rays, with 3p generally comprising records.
%C Records are 3p outside of {1, 2, 4, 12, 18, 24, 25, 30, 36, 42, 56, 70, 105}.
%C a(33) = 35 behaves like a prime; 70 precedes and 105 follows it. a(34) = 105 is conspicuous as it appears earlier than expected. (End)
%H Michael De Vlieger, <a href="/A357963/b357963.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael De Vlieger, <a href="/A357963/a357963.png">Scatterplot of a(n)</a>, n = 1..120, highlighting primes p in red, 2p in blue, and 3p in green.
%H Michael De Vlieger, <a href="/A357963/a357963_1.png">Log-log scatterplot of a(n)</a>, n = 1..2^12, labeling the first 64 terms, showing records in red and local minima in blue, highlighting primes in green and other prime powers in gold. As n increases, the quasi-ray 5*p shows prominently just under the records.
%H Michael De Vlieger, <a href="/A357963/a357963_2.png">Log-log scatterplot of a(n)</a>, n = 1..2^20, demonstrating fine quasi-rays.
%e a(1)=1, a(2)=2 and 2 divides 2 but does not divide 1. Since 2 is the only prime divisor of 2, a(3) = 4, the least unused multiple of 2.
%e Since every prime divisor of a(3)=4 also divides a(2)=2, a(4) = 6, the least novel multiple of the squarefree kernel of 4.
%e a(19), a(20)=25, 30, and 30 has two prime divisors (2,3) which do not divide 25. The least multiples of 2, 3 not seen already are 22, 27 respectively, so a(21)=22.
%e a(29), a(30)=42, 28 and every prime dividing 28 (2,7) also divides 42, so a(31) is 56, the least novel multiple of 14 (squarefree kernel of 28).
%t nn = 68; c[_] = False; q[_] = 1; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; q[2] = 2; Do[m = FactorInteger[a[n - 1]][[All, 1]]; f = Select[m, CoprimeQ[#, a[n - 2]] &]; If[Length[f] == 0, While[Set[k, # q[#]]; c[k], q[#]++] &[Times @@ m], Set[{k, q[#1]}, {#2, #2/#1}] & @@ First@ MinimalBy[Map[{#, Set[g, q[#]]; While[c[g #], g++]; # g} &, f], Last] ]; Set[{a[n], c[k]}, {k, True}], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger_, Oct 22 2022 *)
%Y Cf. A001221, A064413, A352187, A336957.
%K nonn
%O 1,2
%A _David James Sycamore_, Oct 22 2022
%E More terms from _Michael De Vlieger_, Oct 22 2022